Independence complexes of graphs are no less important than other simplicial complexes constructed from graphs and have been studied in many contexts.
In particular, the independence complexes of square grid graphs are studied by Thapper [1]}, Iriye [2]} and many other researchers. It is conjectured by Iriye [2]} that the independence complex of cylindrical square grid graph is always homotopy equivalent to a wedge sum of spheres.
Discrete Morse theory , introduced by Forman [4]} and reformulated by Chari [5]}, is one of the effective methods for determining the homotopy type of independence complex. Bousquet-Mélou, Linusson and Nevo [6]} and Thapper [1]} studied the independence complexes of grid graphs by performing discrete Morse theory as a combinatorial algorithm called matching tree. However, it is hard to distinguish two complexes which has the same number of cells in each dimension only by discrete Morse theory. This is precisely the situation which we have to deal with in this paper. We need topological approaches in case that discrete Morse theory is not available. For example, it is effective to represent an independence complex of a graph as a union of independence complexes of subgraphs, as in Engström [8]}, Adamaszek [9]} and Barmak [10]}.
